A box with a square base and open top must have a volume of 13,500 cm^3. Find the dimensions of the box that minimize the amount of material used.

Question

campbell_st · Answer

ok... so draw a diagram, the base is square.... 

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the volume is 

$$V = x^2 \times h$$

using the given volume 

$$13500 = x^2 \times h$$

make h the subject and 

$$h = \frac{13500}{x^2}$$

now look at the surface area   square base, and 4 rectangular sides all with the same dimensions

$$SA = x^2 + 4(x \times h)$$

makeing the substitution for the height h from the volume formula 

$$SA = x^2 + 4(x \times \frac{13500}{x^2})$$

which will finally give a surface area formula in terms of x as 

$$SA = x^2 + \frac{54000}{x}$$

now the rest should be easy

find the 1st derivative and then solve for x. 

etc... 

hope it makes sense